Linear semi-openness and the Lyusternik theorem

Abstract

A class of set-valued mappings called linearly semi-open mappings is introduced which properly contains the class of linearly open set-valued mappings. A stability result for linearly semi-open mappings is established. The main result is a Lyusternik type theorem.

Sufficient conditions for linear semi-openness of processes are derived. To verify these conditions, the openness bounds of certain processes are computed. A representation of the openness bound of a locally Lipschitz function in a Clarke non-critical point is given.

It is shown that continuous piecewise linear functions on $\text{R}{}^{\mathrm{n}}$ are linearly semi-open under certain algebraic conditions.

Introduction

A classical theorem of Lyusternik [11], modified later by Graves [6], states that the tangent space to a level set of a C¹-mapping F (between Banach spaces) at a point x̄ is the kernel of the Fréchet derivative $\text{F}{}^{\mathrm{\prime }}\text{(}\text{x}\text{̄}\text{)}$––provided the latter is surjective. This hypothesis implies, by Banach's open mapping theorem, that the continuous linear mapping $\text{F}{}^{\mathrm{\prime }}\text{(}\text{x}\text{̄}\text{)}$ is open, which is the crucial property for verifying the statement.

By now, numerous modifications and generalizations of the Lyusternik theorem have been established, where the “derivative” is no longer assumed to be an open linear mapping (see, e.g., [2], [5], [10], [13], [14]). In this context, it turned out that the concept of linear openness (or openness at a linear rate) is an appropriate tool. This concept as well as the closely related metric regularity have been investigated in a lot of papers (we refer in the following to [7], [8], [12], [15]).

The aim of the present paper is to derive a Lyusternik type theorem for a class of set-valued mappings called here linearly semi-open mappings (Section 3). This new concept is equivalently characterized by notions that have their analogies in metric regularity and the pseudo-Lipschitz property (Section 2). Furthermore, Section 2 contains a perturbation result for linear semi-openness. The mappings that are open at a linear rate form a subclass of the linearly semi-open mappings. To provide further examples for linear semi-open mappings, linear processes are investigated (Section 4). In this connection, the computation of openness bounds of linear open mappings in the classical sense gains importance; some results concerning this problem are presented. Finally, conditions are given which ensure that a continuous piecewise linear function is linearly semi-open (Section 5).

Throughout the paper we use the following notations. X and Y are real Banach spaces, B_X and B_Y, or simply B, denotes the respective closed unit ball in the space.

Let $\text{F:X}\text{⇉}\text{Y}$ be a set-valued mapping of X into Y, i.e., a mapping of X into the power set of Y. The set $\text{dom}\text{F={x∈X:F(x)≠∅}}$ is called domain of F. The graph of F is the set gr(F)={(x,y)∈X×Y:y∈F(x)}. We define the kernel of F as the set $\text{ker}\text{F={x∈X:0∈F(x)}}$. The inverse mapping $\text{F}{}^{\mathrm{-1}}\text{:Y}\text{⇉}\text{X}$ is defined by F⁻¹(y)={x∈X:y∈F(x)}. We identify the (single-valued) mapping F:X→Y with the set-valued mapping $\text{F}\text{:X}\text{⇉}\text{Y}$ defined by $\text{F}\text{(x)={F(x)}}$, x∈X.

The closure, the interior, and the convex hull of a subset M of a Banach space are denoted by cl(M), int(M), and co(M), respectively.

A nonempty subset M of a Banach space is said to be a cone, if x∈M and λ⩾0 imply λx∈M. The cone generated by a nonempty set S is denoted by cone(S).